When analyzing potential flow, the Bernoulli equation is indeed commonly used along streamlines to determine the pressure field. The Bernoulli equation relates the pressure, velocity, and elevation of a fluid along a streamline in an inviscid and incompressible flow. It assumes that the fluid particles follow the streamlines, which are paths that are tangent to the velocity vector at every point.
However, if we were to consider the Bernoulli equation across streamlines, it would imply that fluid particles can move between different streamlines, which is not the case in potential flow. In potential flow, streamlines cannot cross or intersect each other, and fluid particles are restricted to following a single streamline.
The absence of streamline crossing is a fundamental characteristic of potential flow, and it ensures that fluid particle paths do not exhibit curvature. Curvature of fluid particle paths would indicate the presence of vorticity and rotational flow, which is not captured by potential flow assumptions. In potential flow, the velocity field is irrotational, meaning that the curl of the velocity vector is zero everywhere.
By using the Bernoulli equation along streamlines, we maintain the assumption of potential flow and ensure that the fluid particle paths are not curved. This allows us to simplify the analysis and obtain useful solutions for various fluid flow problems, such as flow around objects or within certain geometries. However, it's important to note that potential flow is an idealized approximation and may not capture all the details of real-world flows, especially in situations involving vortices or regions of high fluid turbulence.